Linear unmixing of hyperspectral images has rapidly become one of the most widely utilized tools for analyzing the content of hyperspectral images captured by state-of-the-art remote hyperspectral sensors. The aforementioned unmixing process consists of the following three sequential steps: dimensionality estimation, endmember extraction and abundances computation. Within this procedure, the first two steps are by far the most demanding from a computational point of view, since they involve a large amount of matrix operations. Moreover, the complex nature of these operations seriously difficult the hardware implementation of these two unmixing steps, leading to non-optimized implementations which are not able to satisfy the strict delay requirements imposed by those applications under real-time or near real-time requirements. This paper uncovers a new algorithm which is capable of estimating the number of endmembers and extracting them from a given hyperspectral image with at least the same accuracy than state-of-the-art approaches while demanding a much lower computational effort, with independence of the characteristics of the image under analysis. In particular, the proposed algorithm is based on the concept of orthogonal projections and allows performing the estimation of the number of end- members and their extraction simultaneously, using simple operations, which can be also easily parallelized. In this sense, it is worth to mention that our algorithm does not perform complex matrix operations, such as the inverse of a matrix or the extraction of eigenvalues and eigenvectors, which makes easier its ulterior hardware. The experimental results obtained with synthetic and real hyperspectral images demonstrate that the accuracy obtained with the proposed algorithm when estimating the number of endmembers and extracting them is similar or better than the one provided by well-known state-of-the-art algorithms, while the complexity of the overall process is significantly reduced.